FIITJEE CPP - Ellipse

8/20/2019 FIITJEE CPP - Ellipse

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EL

LEVEL-I

1. If equation of ellipse is 16x2 + 25y2 = 400, then eccentricity of the ellipse(A) 2/5 (B) 4/5

(C) 3/5 (D) 1/5

2. If any tangent to the ellipse is2 2

2 2

x y1

a b intercepts lengths h and k on the axes,

then

(A)2 2

2 2

h k1

a b (B)

2 2

2 2

h k2

a b

(C)2 2

2 2

a b1

h k (D)

2 2

2 2

a b2

h k

3. Two perpendicular tangents drawn to the ellipse 116

y

25

x 22

intersect on the curve(A) x= 4 (B) y = 5(C) x2 +y2 = 41 (D) x2 +y2 = 9

4. Equation to an ellipse whose centre is (-2, 3) and whose semi-axes are 3 and 2 andmajor axis parallelto x-axis, is given by

(A) 4x2 + 9y2 + 16 x – 54y – 61 = 0 (B) 4x2 + 9y2 - 16 x + 54y + 61 = 0(C) 4x2 + 9y2 + 16 x – 54y + 61 = 0 (D) none of these

5. The angle between the tangents drawn from the point (1, 2) to the ellipse 3x2 + 2y2 = 5

is(A) tan-1

12

5

(B) tan

-16

5

(C) tan-112

5

(D) tan

-16

5

6. Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from thecentre of theellipse is

(A)3

or 4 4

(B)

2or

3 3

(C)

5or

6 6

(D) none

of these

7. If latus rectum of the ellipse x2 tan2 + y2 sec2 = 1 is 1/2 then ( 0 < ) is

(A)12

(B)

6

(C)

8

12

(D) none of

these


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8. Equation of the ellipse whose minor axis is equal to the distance between foci andwhose latus rectum is 10, is given by(A) 2x2 + 3y2 = 100 (B) 2x2 + 3y2 = 80(C) x2 + 2y2 = 100 (D) none of these

9. If P is a point on the ellipse

2 2x y

16 20 =1 whose foci are S and S. Then PS + PS is

(A) 8 (B) 4 5

(C) 10 (D) 4

10. The distance between the directrices of the ellipse2 2x y

14 9

is

(A)9

5 (B)

24

5

(C)18

5 (D) none of these

11. If F1 (0,0), F2 (3,4) and |PF1| +|PF2| =10, then the locus of is(A) An ellipse (B) A straight line(C) A hyperbola (D) A line segment

12. The locus of a point represented by x =a t 1

2 t

, y =a t 1

2 t

is

(A) an ellipse (B) a circle(C) a pair of line (D) none of these

13. The eccentricity of the conic 7x2 + 16y2 = 112 is

(A) 237

(B) – 34

(C)3

4 (D) none of these

14. The area of the ellipse2 2x y

116 25

is

(A) 16 (B) 20 (C) 25 (D) 36

15. The locus of the point (3h+2, k), where (h, k) lies on the circle x2+y2 = 1 is(A) a hyperbola (B) a circle

(C) a parabola (D) an ellipse

16. The equation of ellipse, whose focus is (1, 0), directrix is x = 4 and whose eccentricity isa root of the quadratic equation 2x2 3x + 1 = 0, is-

(A)2 2x y

14 3

(B)2 2x y

13 4

(C) 4x2 + 3y2 = 24 (D) None of these


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17. Area of the quadrilateral formed by the tangents to the ellipse2

2x y 14

at the end

points of its major and minor axes is(A) 8 (B) 4(C) 16 (D) 2

18. The centre of the ellipse 3x2 + 6x + 4y2 8y 5 = 0, is(A) (1, 1) (B) (1, 1)(C) (1, 1) (D) None of these

19. Length of major axis of the ellipse, 3x2 6x + 4y2 8y 5 = 0, is(A) 4 (B) 1

(C) 3 (D) 2

20. Length of minor axis of the ellipse, 3x2 6x + 4y2 8y 5 = 0, is(A) 4 (B) 2

(C) 3 (D) 2 3


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LEVEL-II

1. The equation2 2x y

1 02 r r 5

represents an ellipse only if

(A) r > 2 (B) r < 5(C) 2 < r < 5 (D) none of these

2. If any tangent to the ellipse2 2x y

116 9

intercepts equal length l on the axes, then l

equals to(A) 25 (B) 7(C) 12 (D) 5

3. An ellipse has its axes along co-ordinate axes. The distance between its foci is 2h andthe focal distanceof an end of the minor axis is k. The equation of the ellipse is

(A)2 2

2 2

x y1

h k

(B)2 2

2 2 2

x y1

k k h

(C)2 2

2 2 2

x y1

k k h

(D)

2 2

2 2 2

x y1

k h h

4. Equation of the ellipse, referred to its axes as the x and y axes respectively, which

passes through the point (-3, 1) and the eccentricity2

5 is

(A) 2x2 + 14y2 = 32 (B) 3x2 + 5y2 = 32(C) 4x2 + 3y2 = 39 (D) none of these

5. Equation of tangents to the ellipse 9x2 + 10y2 = 144 from the point (2, 3) are

(A) y = 3, x + y = 5 (B) x = 3, x – y = 5(C) x + y = 3, x – y + 5 = 0 (D) none of these

6. If a tangent of slope ‘m’ at a point of the ellipse2 2

2 2

x y

a b = 1 passes through (2a, 0) and

if ‘e’ denotes the eccentricity of the ellipse then(A) m2 + e2 = 1 (B) 2m2 + e2 = 1(C) 3m2 + e2 = 1 (D) none of these

7. The tangent to the curve x = a( – sin ); y = a(1 + cos ) at the points = (2k + 1), k Z are parallel to(A) y = x (B) y = –x

(C) y = 0 (D) x = 0

8. The equation(s) of the tangent(s) to the ellipse 9(x - 1)2 + 4y2 = 36 parallel to the latusrectum, is (are)(A) y = 3 (B) y = -3(C) x = 3 (D) x = -3.

9. The area of the triangle formed by the points on the ellipse 25x2 + 16y2 = 400 whoseeccentric angles are /2, and 3/2 is


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(A) 10 sq. units (B) 20 sq. units(C) 30 sq. units (D) 40 sq. units

10. If 3 bx+ay = 2ab touched the ellipse2 2

2 2

x y

a b = 1 then eccentric angle is

(A) 6

(B) 4

(C)3

(D)

2

11. The vaue of ‘c’ for which lie y = x + c is tangent to the ellipse 2x2 + 3y2 = 1 is

(A)6

7 (B)

5

6

(C)2

3 (D)

3

2

12. Foci of the ellipse; 25 (x + 1)2

+9(y +2)2

= 225, are(A) (1, 2) and (1, 6) (B) (2, 1) and (2, 6)(C) (1, 2) and (1, 6) (D) (1, 2) and (1, 6)

13. Let ‘E’ be the ellipse2 2x y

19 4

and ‘C’ be the circle x2 + y2 = 9. Let P and Q be points

(1, 2) and (2, 1) respectively. Then(A) ‘Q’ lies in side ‘C’ but outside E (B) ‘Q’ lies outside both C and E(C) P lies inside both C and E (D) P lies inside ‘C’ but outside E

14. The equation 3 (x + y 5)2 + 2 (x y + 7)2 = 6 represents an ellipse, whose centre is(A) (1, 6) (B) (6, 1)(C) (1, 6) (D) (6, 1)

15. Eccentricity of the ellipse 3 (x + y 5)2 + 2 (x y + 7)2 = 6 is

(A)1

2 (B)

2

3

(C)1

3 (D)

1

2

16. One foot of normal of the ellipse 4 x2 + 9y2 = 36, that is parallel to the line 2 x + y = 3, is

(A)9 8

,

5 5

(B)9 8

,

5 5

(C)9 8

,5 5

(D) None of these

17. Equation of the ellipse whose axes are coordinate axes and whose length of latusrectum, and eccentricity are equal and equal to ½ each is(A) 6 x2 + 12 y2 = 1 (B) 12 x2 + 6 y2 = 1(C) 3 x2 + 12 y2 = 1 (D) 9 x2 + 12 y2 = 1


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18. The line y = x 1 touches the ellipse 3 x2 + 4 y2 = 12, at

(A) (1/2, 1/2) (B) (3, 2)(C) (1, 2) (D) None of these

19. The equation of common tangents to the curves x2 + 4 y2 = 8 and y2 = 4 x are(A) 2 y x 4 = 0 and 2 y + x + 4 = 0 (B) y 2 x 4 = 0 and y + 2 x + 4 = 0(C) 2 y x 2 = 0 and 2 y + x + 2 = 0 (D) y 2 x 2 = 0 and y + 2 x + 2 = 0

20. If the line y = mx + c is a tangent to the ellipse2 2

2 2

x y1

a b then corresponding point of

contact is

(A)2 2a m b

,c c

(B)2 2a m b

,c c

(C)2 2a m b

,c c

(D)2 2a m b

,c c

LEVEL-III

1. The length of the major axis of the ellipse2

2 2 (3x 4y 7)(5x 10) (5y 15)4

is

(A) 10 (B) 20/3(C) 20/7 (D) 4

2. An ellipse has eccentricity 1/2 and one focus at the point P(1/2, 1). One of Its

directrix is the common tangent, nearer to the point P to the circle x2+y2 = 1 andthe hyperbola x2-y2 = 1. Area of the ellipse is

(A) (B)2 2

(C)2

3 3

(D) none of these.

3. If the normal at the point P() to the ellipse2 2x y

114 5

intersects it again at the point

Q(2), then cos =(A) 2/3 (B) -2/3 (C) 1/3 (D) -1/3

4. The equation of the ellipse centered at (1, 2) having the point (6, 2) as one of its focusand passing through the point (4, 6) is

(A)

2 2

x 1 3 y 21

36 64

(B)

2 2

x 1 y 21

45 20

(C)

2 2

x 1 y 21

18 32

(D)

2 2

x 1 7 y 21

72 128


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5. The tangent drawn to the ellipse2x 11y

116 256

at the point P (); touches the circle

(x 1)2 + y2 = 16; then ‘’ equal to(A) /6 (B) /4

(C) /3 (D) None of these

6. Length of latus rectum of the ellipse, 3 (x + y 5)2 + 2 (x y + 7)2 = 6 is

(A)2

43

(B)2

23

(C)1

3 (D)

2

3

7. Focii of the ellipse; 3 (x + y 5)2 + 2 (x - y + 7)2 = 6 are(A) (1/2, 13/2) and (3/2, 11/2) (B) (1/2, 11/2) and (3/2, 13/2)(C) (1/2, 11/2) and (-3/2, 11/2) (D) (1/2, 11/2) and (3/2, 13/2)

8. Locus of the midpoint of chords of the ellipse2 2

2 2

x y1

a b that are parallel to the line

y = 2 x + c, is(A) 2 b2 y a2 x = 0 (B) 2 a2 y b2 x = 0(C) 2 b2 y + a2 x = 0 (D) 2 a2 y + b2 x = 0

9. Consider an ellipse2 2

2 2

x y1

a b , centered at point ‘O’ and having AB and CD as its major

and minor axes respectively. If S 1 be one of the focus of the ellipse, radius of incircle oftriangle OCS1 = 6 units, then area of the ellipse is equal to

(A) 16 sq. units (B) 654 sq. units

(C)65

2sq. uints (D) 65 sq. units

10. ‘P’ is any variable point on the ellipse2 2

2 2

x y1

a b having the points S1 and S2 as its foci .

maximum area of the triangle PS1S2 is(A) b2 c sq. units (B) a2 c sq. units(C) ab sq. units (D) abc sq. units

11. Consider an ellipse having its axes as coordinate axes and passing through the point(4, 1). If the line x +4y 10 = 0 is one of its tangent, then area of ellipse is(A) 10 (B) 20 (C) 25 (D) 15


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12. S1 and S2 are foci of an ellipse ‘B’ be one of the extremity of its minor axes. If S1 S2 Bis right angled then eccentricity of the ellipse is equal to

(A)3

2 (B)

1

2

(C)

3

2 (D) None of these

13. If ‘L’ is the length of perpendicular drawn from the origin to any normal of the ellipse2 2x y

1,25 16

then maximum value of ‘L’ is

(A) 5 (B) 4(C) 1 (D) None of these

14. The maximum distance of the centre of the ellipse2 2x y

19 4

from the chord of contact

of mutually perpendicular tangents of the ellipse is

(A) 913

(B) 313

(C)6

13 (D)

36

13

15. Tangents PA and PB are drawn to the ellipse2 2x y

116 9

from the point P (0, 5). Area of

triangle PAB is

(A)16

5 (B)

256

25

(C)35

2 (D)1024

25

16. The straight line x 2y + 4 = 0 is one of the common tangents of the parabola y2 = 4x

and2 2

2

x y1

4 b . The equation of the another common tangent to these curves is

(A) x + 2y + 4 = 0 (B) x + 2y 4 = 0(C) x + 2y + 2 = 0 (D) x + 2y 2 = 0

17. A variable tangent of the ellipse2 2x y

116 36

meets the tangents drawn at the extremities

of the major axis at point A1 and A2 Circle drawn on A1 A2 as diameter will always pass

through two fixed points whose coordinates are(A) (0, 6) (B) (0, 5 2 )

(C) (0, 2 5 ) (D) (0, 4)


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18. There are exactly two points on the ellipse2 2

2 2

x y1

a b whose distance from the center of

the ellipse are equal and equal to2 2a b

2

. Eccentricity of this ellipse is

(A)

3

2 (B)

2

3

(C)1

3 (D)

2

3

19. For all admissible values of the parameter ‘a’ the straight line 2 ax + y 21 a 1 willtouch an ellipse whose eccentricity is equal to

(A)3

2 (B)

2

3

(C)3

2 (D)

2

3

20. The normal to the ellipse 4 x2 + 5 y2 = 20 at the point ‘P’ touches the parabola y2 = 4x,the eccentric angle of ‘P’ is

(A) 11

sin5

(B) 11

tan2 5

(C) 11

tan5

(D) 11

cos5


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ANSWERS

LEVEL −I

1. C 2. C 3. C 4. C5. 6. C 7. A 8. C9. B 10. C 11. A 12. D13. C 14. B 15. D 16. A

17. A 18. C 19. A 20. D

LEVEL −II

1. C 2. D 3. C 4. B5. D 6. C 7. C 8. A9. B 10. A 11. B 12. C13. D 14. A 15. C 16. C17. D 18. D 19. A 20. D

LEVEL −III

1. B 2. D 3. B 4. B5. C 6. B 7. A 8. D

9. B 10. A 11. A 12. B13. C 14. A 15. B 16. A17. C 18. D 19. A 20. D

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FIITJEE CPP - Ellipse